308 Prof. L. Boltzmann on the Assumptions necessary 



The right-hand side agrees completely with the expression 



P 



which Prof. Tait finds on p. 346 for — (u' 2 — u 2 ), since, 



according to equation (20) of my treatise, the quantity which 

 Prof. Tait denotes by u has the value v(—gcr + <yso), where 

 u — v is the projection of r on OC, which is equal to rtr. Now 

 let there be very many molecules of mass m (molecules of the 

 first kind), as also of those of mass M (molecules of the second 

 kind) } uniformly mixed in a space and uniformly distributive 

 with reference to the directions of their velocities. Of all 

 possible impacts which in general may occur, we will first 

 select only those for which the variables v, V, T, S, lie 

 between the infinitely close limits, 



v&ndv + dv, VandV + dV, TandT + dT,l 



SandS + dS, OandO + dO. J ' '() 



In fig 1 a sphere of radius S, supposed concentric with 

 one of the molecules of mass M, is drawn. All straight lines 

 are drawn dotted from their intersections with the surface of 

 this sphere. The arcs are arcs of great circles of this sphere. 

 The molecules of mass m fly against this sphere, so that their 

 centres describe the straight line AC relative to the first 

 molecule which is parallel to vY and OR. At the point C 

 the centres are reflected in the direction CB, which is parallel 

 to v'Y and OR'. The relative motion before impact, with 

 which alone we are concerned, remains unaltered if we imagine 

 the molecules of mass m at rest and the sphere drawn in fig. 1 

 moving with the velocity r = vY in direction opposite to OR. 

 If its surface be divided by a plane passing through its centre 

 at right angles to OR, then, with our last-mentioned con- 

 ception of relative motion, the preceding half-sphere would, 

 in unit time, pass through a space bounded by two half-spheres 

 and a cylindrical surface of volume 7rB 2 r. Of this whole 

 space a small portion is now to be cut out, as follows : — OC 

 is that radius of the sphere which is parallel to the direction 

 of the line of centres. Whilst the point C is so moved on the 

 surface of the sphere that S increases by the amount rfS without 

 change of direction of OR, C describes a linear element of 

 length SdS. If, on the other hand, we move C so that 

 increases by dO, C describes a linear element of length SsdO. 

 These two determine an element of area on the surface of the 

 sphere of area 8 2 sd$dO. This is inclined at an angle 8 to the 

 direction of r. Since the half-sphere moves with the velocity 

 r in the direction —OR, this element of surface moves through 

 a prism of volume r$ 2 scr d$ dO. So soon as the centre of a 



